Doléans-Dade exponential

In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential of a complex-valued semimartingale X is defined to be the unique strong solution of the stochastic differential equation

$$\mathrm {d} Y_{t}=Y_{t-}\mathrm {d} X_{t},\qquad {\textsf {with}}\ {\textsf {the}}\ {\textsf {initial}}\ {\textsf {condition}}\ Y_{0}=1.$$

The concept is named after Catherine Doléans-Dade, who studied it in . Process $Y$ obtained above is commonly denoted by ${\mathcal {E}}(X)$. If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation $\mathrm {d} Y_{t}/\mathrm {d} t=Y_{t}\mathrm {d} X_{t}/\mathrm {d} t$, whose solution is $Y=\exp(X-X_{0})$. Alternatively, if X_t = σB_t + μt for a Brownian motion B, then the Doléans-Dade exponential is a geometric Brownian motion. For any continuous semimartingale X, applying Itō's lemma with ƒ(Y) = log(Y) gives

${\begin{aligned}d\log(Y)&={\frac {1}{Y}}\,dY-{\frac {1}{2Y^{2}}}\,d[Y]\\&=dX-{\frac {1}{2}}\,d[X].\end{aligned}}$

Exponentiating gives the solution

$Y_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}{\Bigr )},\qquad t\geq 0.$

This differs from what might be expected by comparison with the case where X is differentiable due to the existence of the quadratic variation term [X] in the solution. Note that the above argument is a heuristic one, as we do not know a priori that a semimartingale solution to the stochastic differential equation exists. Also, the logarithm is not a twice differentiable and continuous function on the real numbers.

The Doléans-Dade exponential is useful in the case when X is a local martingale. Then, Ɛ(X) will also be a local martingale whereas the normal exponential exp(X) is not. This is used in the Girsanov theorem. Criteria for a continuous local martingale X to ensure that its stochastic exponential Ɛ(X) is actually a martingale are given by Kazamaki's condition, Novikov's condition, and Beneš's condition.

It is possible to apply Itō's lemma for non-continuous semimartingales in a similar way to show that the Doléans-Dade exponential of any semimartingale X is

$Y_{t}=\exp {\Bigl (}X_{t}-X_{0}-{\frac {1}{2}}[X]_{t}{\Bigr )}\prod _{s\leq t}(1+\Delta X_{s})\exp {\Bigl (}-\Delta X_{s}+{\frac {1}{2}}\Delta X_{s}^{2}{\Bigr )},\qquad t\geq 0,$

where the product extents over the (countably many) jumps of X up to time t.

See also

• Stochastic logarithm

References

• Jacod, J.; Shiryaev, A. N. (2003), Limit Theorems for Stochastic Processes (2nd ed.), Springer, pp. 58--61, ISBN 3-540-43932-3
• Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 3-540-00313-4